Parameter Tuning for Local-Search-Based Matheuristic Methods

<table class="figure-group"><tr class="fig-image" id="a"><td><object data="https://static.hindawi.com/articles/complexity/volume-2017/1702506/figures/1702506.fig.004a.svgz" name="1702506.fig.004a" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(a) </b>Evolution of the GAP for <svg height="9.49473pt" id="M353" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 34.47 9.49473" width="34.47pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M699 369C699 549 575 666 407 666C186 666 23 488 23 278C23 101 145 -16 312 -16C535 -16 699 153 699 369ZM600 373C600 210 500 19 321 19C186 19 120 129 120 272C120 450 232 631 399 631C541 631 600 522 600 373Z" id="g113-80"></path></g><g transform="matrix(.013,0,0,-0.013,8.996,0)"><path d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z" id="g113-83"></path></g><g transform="matrix(.013,0,0,-0.013,17.147,0)"><path d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z" id="g113-77"></path></g><g transform="matrix(.013,0,0,-0.013,24.593,0)"><path d="M244 607C244 633 228 655 200 655C166 655 146 618 146 594C146 564 166 546 191 546C221 546 244 574 244 607ZM222 91L209 114C184 94 148 66 133 66C127 66 124 73 130 96L201 370C213 416 211 448 191 448C162 448 88 407 29 352L42 328C73 354 104 371 114 371C120 371 119 365 115 345L53 92C32 5 45 -12 68 -12C103 -12 186 50 222 91Z" id="g113-106"></path></g><g transform="matrix(.013,0,0,-0.013,28.142,0)"><path d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z" id="g113-99"></path></g></svg>  <svg height="9.09021pt" id="M354" style="vertical-align:-0.2455397pt" version="1.1" viewbox="-0.0498162 -8.84467 11.2563 9.09021" width="11.2563pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M837 672C829 675 822 674 813 674C710 674 608 541 510 398C475 447 422 480 349 480C260 480 162 415 162 299C162 199 212 133 284 92C229 39 191 15 134 15C91 15 69 45 69 65H71C78 54 95 53 100 53C128 53 143 80 143 101C143 128 121 149 93 149C60 149 31 121 31 77C31 24 74 -15 138 -15C198 -15 255 22 311 79C360 56 416 43 473 39L455 0H552L569 42C612 49 661 63 693 87L684 103C657 88 617 76 581 69C677 301 753 477 846 658L837 672ZM772 638C701 530 628 379 559 227C558 279 547 330 526 371C609 494 688 609 771 638H772ZM536 199C536 191 536 184 535 176L486 67C430 71 380 84 335 105C395 173 451 257 507 344C527 301 536 251 536 199ZM492 372C431 277 369 183 310 119C245 158 203 217 203 301S275 447 349 447C416 447 461 417 492 372Z" id="g198-2"></path></g></svg> instances</td></tr><tr class="fig-image" id="b"><td><object data="https://static.hindawi.com/articles/complexity/volume-2017/1702506/figures/1702506.fig.004b.svgz" name="1702506.fig.004b" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(b) </b>Evolution of the time for <svg height="9.49473pt" id="M355" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 34.47 9.49473" width="34.47pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M699 369C699 549 575 666 407 666C186 666 23 488 23 278C23 101 145 -16 312 -16C535 -16 699 153 699 369ZM600 373C600 210 500 19 321 19C186 19 120 129 120 272C120 450 232 631 399 631C541 631 600 522 600 373Z" id="g113-80"></path></g><g transform="matrix(.013,0,0,-0.013,8.996,0)"><path d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z" id="g113-83"></path></g><g transform="matrix(.013,0,0,-0.013,17.147,0)"><path d="M541 160L512 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35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z" id="g113-99"></path></g></svg>  <svg height="9.09021pt" id="M356" style="vertical-align:-0.2455397pt" version="1.1" viewbox="-0.0498162 -8.84467 11.2563 9.09021" width="11.2563pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M837 672C829 675 822 674 813 674C710 674 608 541 510 398C475 447 422 480 349 480C260 480 162 415 162 299C162 199 212 133 284 92C229 39 191 15 134 15C91 15 69 45 69 65H71C78 54 95 53 100 53C128 53 143 80 143 101C143 128 121 149 93 149C60 149 31 121 31 77C31 24 74 -15 138 -15C198 -15 255 22 311 79C360 56 416 43 473 39L455 0H552L569 42C612 49 661 63 693 87L684 103C657 88 617 76 581 69C677 301 753 477 846 658L837 672ZM772 638C701 530 628 379 559 227C558 279 547 330 526 371C609 494 688 609 771 638H772ZM536 199C536 191 536 184 535 176L486 67C430 71 380 84 335 105C395 173 451 257 507 344C527 301 536 251 536 199ZM492 372C431 277 369 183 310 119C245 158 203 217 203 301S275 447 349 447C416 447 461 417 492 372Z" id="g198-2"></path></g></svg> instances</td></tr></table>

<div>Average results obtained by the local search algorithms for each value of <svg height="11.5564pt" id="M350" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 36.275 11.5564" width="36.275pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path></g><g transform="matrix(.013,0,0,-0.013,13.644,0)"><path d="M535 230V280H52V230H535Z" id="g117-33"></path></g><g transform="matrix(.013,0,0,-0.013,24.18,0)"><path d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z" id="g113-223"></path></g><g transform="matrix(.013,0,0,-0.013,31.565,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path></g></svg> for <svg height="9.49473pt" id="M351" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 34.47 9.49473" width="34.47pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M699 369C699 549 575 666 407 666C186 666 23 488 23 278C23 101 145 -16 312 -16C535 -16 699 153 699 369ZM600 373C600 210 500 19 321 19C186 19 120 129 120 272C120 450 232 631 399 631C541 631 600 522 600 373Z" id="g113-80"></path></g><g transform="matrix(.013,0,0,-0.013,8.996,0)"><path d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z" id="g113-83"></path></g><g transform="matrix(.013,0,0,-0.013,17.147,0)"><path d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z" id="g113-77"></path></g><g transform="matrix(.013,0,0,-0.013,24.593,0)"><path d="M244 607C244 633 228 655 200 655C166 655 146 618 146 594C146 564 166 546 191 546C221 546 244 574 244 607ZM222 91L209 114C184 94 148 66 133 66C127 66 124 73 130 96L201 370C213 416 211 448 191 448C162 448 88 407 29 352L42 328C73 354 104 371 114 371C120 371 119 365 115 345L53 92C32 5 45 -12 68 -12C103 -12 186 50 222 91Z" id="g113-106"></path></g><g transform="matrix(.013,0,0,-0.013,28.142,0)"><path d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z" id="g113-99"></path></g></svg>  <svg height="9.09021pt" id="M352" style="vertical-align:-0.2455397pt" version="1.1" viewbox="-0.0498162 -8.84467 11.2563 9.09021" width="11.2563pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M837 672C829 675 822 674 813 674C710 674 608 541 510 398C475 447 422 480 349 480C260 480 162 415 162 299C162 199 212 133 284 92C229 39 191 15 134 15C91 15 69 45 69 65H71C78 54 95 53 100 53C128 53 143 80 143 101C143 128 121 149 93 149C60 149 31 121 31 77C31 24 74 -15 138 -15C198 -15 255 22 311 79C360 56 416 43 473 39L455 0H552L569 42C612 49 661 63 693 87L684 103C657 88 617 76 581 69C677 301 753 477 846 658L837 672ZM772 638C701 530 628 379 559 227C558 279 547 330 526 371C609 494 688 609 771 638H772ZM536 199C536 191 536 184 535 176L486 67C430 71 380 84 335 105C395 173 451 257 507 344C527 301 536 251 536 199ZM492 372C431 277 369 183 310 119C245 158 203 217 203 301S275 447 349 447C416 447 461 417 492 372Z" id="g198-2"></path></g></svg> instances.</div>

Complexity

Parameter Tuning for Local-Search-Based Matheuristic Methods

Figure 4